What exactly do we mean by equilibrium constants? Isn't there just one? Well, technically there is only one equilibrium position, but we can define multiple equilibrium constants to characterize it.
For a more detailed explanation, read about the equilibrium constants , but we generally express them in terms of:
We've written the equilibrium constants for the general reaction \(\ce{aA + bB <=> cC + dD}\)
In general, we only use \(K_x\) and \(K_p\) for gas phase reactions.
It is known that the partial pressure of a gas is defined as \(p = x\cdot P\), where P is the total pressure.
We have:
\(K_p = \frac{p_C^cp_D^d}{p_A^ap_B^b} = \frac{(x_CP)^c(x_DP)^d}{(x_AP)^a(x_BP)^b} = \frac{x_C^cx_D^d}{x_A^ax_B^b}\cdot P^{c+d-a-b} = K_x \cdot P^{\Delta n}\)
We denoted the number of moles variation per reaction as \(\Delta n\)
For a certain gas phase substance, the molar concentration is \(c = \frac{n}{V} = \frac{p}{RT}\), from the ideal gas law.
Similar to before (try to complete the proof yourself if you're unsure), we have \(K_c = K_p \cdot (RT)^{-\Delta n}\)
\(K_c = K_p \cdot (RT)^{-\Delta n} = K_x \cdot P^{\Delta n} \cdot (RT)^{-\Delta n} = K_x \cdot (\frac{P}{RT})^{\Delta n}\)
Later in your olympiad student career you will learn that equilibrium constants are all written in terms of the same (dimensionless) quantity, called activity.
For solutes, the activity is approximately equal to the ratio between their concentration and the standard 1M concentration. For gases, it's roughly the same as the ratio between the partial pressure and the standard 1atm pressure. But it's always the same thing.
One thing that confuses a lot of people is the unit. Sometimes we wonder how we can take the log of something that has a unit (you will learn the formula \(\Delta G = -RT\ln (K)\)). Sometimes it feels weird to use pressure for gases and concentrations for solutes when writing the Nernst equation. For all of these questions, the answer is the same - you are using the activity of a compound and only applying appropriate formulas each time.
Regardless of al this, as shown by the formulas above, in some cases, we can express equilibrium constants in terms of whatever is more useful. It's all up to what you feel would work best mathematically.
Written by Alex Jicu